The Knotted String

We've seen a paper on the Thrown String. Here is one on the Knotted String.University of California at San Diego physicists Raymer and Smith place various lengths of string in a box and film it tumbling for ten seconds. More specifically from their PNAS paper "Spontaneous knotting of an agitated string,"

Most of the measurements were carried out with a string having a diameter of 3.2 mm, a density of 0.04 g/cm, and a flexural rigidity of 3.1 × 10⁴ dynes·cm², tumbling in a 0.30 × 0.30 × 0.30-m box rotated at one revolution per second for 10 sec.

Results from 200 trials noted the proportion of knots formed for various lengths. These results are plotted above. The dependence of this knotted probability on other physical properties of the string are shown in their table below:

They conjecture that the string confinement and rotation promote braiding at the ends of the string, producing the knots. As one report noted  Apple's iPhone earbuds are 139 cm (55 inches) long and thus right at the 50% tangle-rate-sweet-spot at the top of the curve. Shorter earbuds would be welcome.